Is there any way to prevent lights in linked files from behaving very differently?
Instanced light don't cast shadows in 2.80, EEVEE doesn't support casting shadows from duplis (particles, collection instances, duplifaces, etc.). It is a know limitation of the current system.

In 2.82 and forward the shadow system has been reworked to allow this, so either wait it out, try the risks of using an experimental version, or you will have to make your light setup local.